As the apartment market soars across the U.S., one financial-services startup sees opportunity in helping tenants get into units they can’t qualify for on their own.

TheGuarantors, launched in New York in 2014, sells payment insurance to tenants, providing landlords with a guarantee they will be made whole if the tenant becomes delinquent.

The offering is a symptom of a pricey market in which rents have risen faster than incomes and landlords can be picky about the qualifications they demand. Rents have climbed about 20% nationwide over the past five years while incomes have only recently started to rise.

The insurance, similar to the private mortgage insurance many lenders require of borrowers who have small down payments, provides a new level of protection for developers putting up new buildings in pricey markets where the applicant pool might be thin.

It could be a boon to landlords in places like San Francisco and New York in particular because rent growth has far outstripped income gains in recent years. Cliff Finn,executive vice president of new development at Douglas Elliman in New York, said 10% to 30% of the tenants in buildings he is leasing are now insured through TheGuarantors.

Some of New York City’s largest apartment complexes, from the 11,200-unit Stuyvesant Town and Peter Cooper Village on Manhattan’s East Side to the 1,258-unit Gotham West complex near Hudson Yards, are willing to accept TheGuarantors, the company said.

“It reduces the exposure to vacancy,” said Jonathan Miller, president of appraisal firm Miller Samuel, who sits on TheGuarantors’s advisory board. “It basically opens it to another category of tenant.”

In New York, landlords have stiff requirements. They typically demand pay stubs showing that prospective tenants’ annual income is at least 40 times greater than one month’s rent. Tenants also must have credit scores of about 700 or better on a scale of 300 to 850 and solid rental histories. Those who can’t meet those requirements must find a guarantor who earns 80 times the monthly rent—no easy feat as rents keep rising.

TheGuarantors has a lower bar. It insures tenants who earn as little as 27 times the monthly rent and have credit scores as low as 630. The company also takes into account savings and other liquid assets and income earned outside the country. Even the cost and quality of degree a college student is pursuing can offer clues as to how likely they are to miss a rent payment.

The premiums work out to two weeks to about a month’s worth of rent a year, depending on how risky the applicant is. The company, which issued its first policy earlier this year, said it is experiencing its first default.

“We’re taking a little more risk than the landlord, but we’re charging a premium for that,” said Julien Bonneville, founder and chief executive of TheGuarantors.

The company is partnering with Worcester, Mass.-based Hanover Insurance Group Inc., a $5 billion insurance company founded 160 years ago.

For tenants, the product is likely to produce mixed results. It will help some people who otherwise wouldn’t qualify for an apartment. But the worry is that landlords could decide to require the insurance from borderline applicants who might have qualified for the apartment anyway.

Mr. Bonneville, a native of France, moved to New York in 2010 to attend business school at Columbia University. As a student with little income, he struggled to qualify for an apartment. He eventually used the services of Insurent, which was launched in 2008 to offer a similar type of insurance to tenants.

Mr. Bonneville formed TheGuarantors in 2014 and said he now has about 300 buildings signed up. He declined to disclose how many guarantees the company has issued, but said it is profitable.

Insurent remains the biggest player in the nascent market. The company said it is accepted at 3,000 buildings and has issued some 13,000 guarantees over the past eight years.

Xiao Xu, a 25-year-old recent graduate of the Illinois Institute of Technology, likely wouldn’t have been able to find an apartment without such insurance. Mr. Xu is starting an internship at a New York bank in October but needed an apartment this month. He found a one-bedroom apartment on the Upper East Side for about $3,500 a month, but his starting salary of $20,000 was nowhere near enough.

He said he contacted TheGuarantors at midnight and got a reply the next morning. Everything was completed the next afternoon, he said.

Without getting insurance, qualifying for the apartment would have been “impossible,” he said.